00000133.htm

116 VI HUBERT SPACES

expresses that /is symmetric. For any finite systems (x_f), (j^-), (a_£), (jfy), we
have

(6.1.1)

i J

by induction on the number of elements of these systems.

From (6.1.1) it follows that if E is finite dimensional and (a_t) is a basis
of E, /is entirely determined by its values a_fj- =f(a_iy 0/), which are such
that (by V))

(6.1.2) o^fiy.

Indeed we have then, for x = £ ^a,., j> = £ 1^0,-

i i

(6.1-3)

Conversely, for any system (a_l7) of real (resp. complex) numbers satisfying
(6.1.2), the right hand side of (6.1.3) defines on the real (resp. complex)
finite dimensional vector space E a hermitian form.

Example

(6.1.4) Let D be a relatively compact open set in R², and let E be the real
(resp. complex) vector space of all real-valued (resp. complex- valued) bounded
continuous functions in D, which have bounded continuous first derivatives
in D. Then the mapping

, 9) - <f(f, 9) = 1 1 | t(x, )>)/O, rt»(x, y)

(where a, b, c are continuous, bounded and real-valued in D) is a hermitian
form on E.

A pair of vectors X₉y of a vector space E is orthogonal with respect to
a hermitian form f on E if f(x, y) = 0 (it follows from (V) that the relation
is symmetric in x, y) ; a vector x which is orthogonal to itself (i.e./(x, x) = 0)
is isotropic with respect to /. For any subset M of E, the set of vectors y
which are orthogonal to all vectors x e M is a vector subspace of E, which
is said to be orthogonal to M (with respect to /). It may happen that there
exists a vector a ^ 0 which is orthogonal to the whole space E, in which
case we say the form / is degenerate. On a finite dimensional space E, non-
degenerate hermitian forms / defined by (6.1.3) are those for which the
matrix (a_l7) is invertible.



	116 VI HUBERT SPACES expresses that /is symmetric. For any finite systems (x_f), (j^-), (a_£), (jfy), we have

	(6.1.1) i J by induction on the number of elements of these systems. From (6.1.1) it follows that if E is finite dimensional and (a_t) is a basis of E, /is entirely determined by its values a_fj- =f(a_iy 0/), which are such that (by V)) (6.1.2) o^fiy. Indeed we have then, for x = £ ^a,., j> = £ 1^0,- i i (6.1-3)

	Conversely, for any system (a_l7) of real (resp. complex) numbers satisfying (6.1.2), the right hand side of (6.1.3) defines on the real (resp. complex) finite dimensional vector space E a hermitian form. Example (6.1.4) Let D be a relatively compact open set in R², and let E be the real (resp. complex) vector space of all real-valued (resp. complex- valued) bounded continuous functions in D, which have bounded continuous first derivatives in D. Then the mapping


	, 9) - <f(f, 9) = 1 1 \| t(x, )>)/O, rt»(x, y)


	(where a, b, c are continuous, bounded and real-valued in D) is a hermitian form on E. A pair of vectors X₉y of a vector space E is orthogonal with respect to a hermitian form f on E if f(x, y) = 0 (it follows from (V) that the relation is symmetric in x, y) ; a vector x which is orthogonal to itself (i.e./(x, x) = 0) is isotropic with respect to /. For any subset M of E, the set of vectors y which are orthogonal to all vectors x e M is a vector subspace of E, which is said to be orthogonal to M (with respect to /). It may happen that there exists a vector a ^ 0 which is orthogonal to the whole space E, in which case we say the form / is degenerate. On a finite dimensional space E, non- degenerate hermitian forms / defined by (6.1.3) are those for which the matrix (a_l7) is invertible.